Problem: Find one value of $x$ that is a solution to the equation: $(3x+7)^2=-6x-14$ $x=$
Explanation: We could solve for $x$ by expanding $(3x+7)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $-6x-14=-2({3x+7})$. This means that we can rewrite the equation as: $({3x+7})^2=-2({3x+7})$ If we let ${p}={3x+7}$, we can see that this equation is in the form: ${p}^2=-2{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2&=-2{p}\\\\ {p}^2+2{p}&=0\\\\ {p}({p}+2)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=-2 \end{aligned}$ Since ${p}={3x+7}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${3x+7}=0\ \ \ \text{or} \ \ \ {3x+7}=-2$ When we solve ${3x+7}=0$, we find that $x=-\dfrac{7}{3}$. When we solve ${3x+7}=-2$, we find that $x=-3$. In conclusion, the two solutions of the equation $(3x+7)^2=-6x-14$ are $x=-\dfrac{7}{3}$ and $x=-3$. [Is there another way to solve for x?]